Floor, not round
Put a package of length L on rollers spaced P apart. Depending on where its leading edge lands relative to the nearest roller, the number of rollers underneath is either the floor of L over P or one more than that. It is never fewer and never more, and as the package travels it passes through both states.
A 10 in carton on 3 in centres gives 10 divided by 3, which is 3.33, so the worst case is three rollers and the best is four. Rounding 3.33 to the nearest whole number gives three here and would give four for a 10.6 in carton, which is a claim about a position the carton only sometimes occupies. Always take the floor.
The same fact from the other end
If your rule is three rollers under the shortest package, then the pitch has to be no more than L divided by three, which is 3.33 in for a 10 in carton. Read the other way: at 3 in centres, the shortest package that always gets three rollers is exactly 9 in. That second number is the useful one, because it is a property of the conveyor you already own and it answers the question a buyer asks when the carton range changes.
It steps, it does not slide
Tightening the pitch from 3 in to 2.6 in changes nothing at all for a 10 in package: the floor of 10 over 2.6 is still three. Going to 2.5 in ticks it to four. Half the pitch reductions people pay for buy no extra support for the package they were worried about, and the only way to know which is which is to do the division against the specific shortest item rather than in general.
Fence posts
A 40 ft run at 3 in centres contains 160 pitches and 161 rollers, because there is a roller at each end of every pitch. Whether the frames you buy are drilled to give you that extra one at a butt joint, or whether two adjacent sections share it, depends on the supplier. Count one section before ordering a run of them.
Where the arithmetic stops
All of this assumes a rigid flat bottom. A polybag settles into the gap, a shrink-wrapped bundle drags, a tote on feet is supported wherever its feet are and not where its outline is, and a pallet with runners across the rollers behaves like a completely different problem. The clear gap between rollers, which is pitch minus roller diameter, is the span the underside has to bridge, and for anything floppy that span is the criterion rather than the roller count. Load sharing between rollers is not even either, so the load-per-roller line is a division and not a rating.
Questions people ask
How many rollers should be under a box on a roller conveyor?
That is a rule you or your conveyor supplier set for your packages, not something arithmetic decides. Three under the shortest rigid flat-bottomed carton is what a lot of shops work to. What this page does is turn whatever rule you pick into a pitch, and turn whatever pitch you have into the shortest package it satisfies.
How do I calculate roller spacing for a conveyor?
Divide the shortest package length by the number of rollers you want under it. A 10 in package with a three roller rule gives 3.33 in maximum pitch. Going the other way, at 3 in centres the shortest package that always sees three rollers is 9 in, since three times the pitch is exactly the length at which the floor division reaches three.
Why round down instead of rounding normally?
Because the package lands wherever it lands. A 10 in package on 3 in centres has four rollers under it at some positions and three at others, and it moves through both continuously. The three is the one that decides whether it stalls or drops a corner, so the floor is the honest answer and rounding to nearest describes a position the package is not always in.
How many rollers are in a 40 foot run at 3 inch centres?
161. There are 160 pitches of 3 in in 480 in, and a roller sits at each end of every pitch, so the count is one more than the number of gaps. Whether adjacent frame sections share the roller at a butt joint varies by supplier, so count one section from the drawing before multiplying it out.
Does this work for polybags and totes?
Not really, and the page says so rather than pretending. Roller count assumes a rigid flat bottom that bridges the gaps. A polybag settles into the gap between rollers and the governing dimension becomes that clear gap, pitch minus roller diameter. A tote or pallet on feet or runners is supported where its feet are, which has nothing to do with its overall length. Those cases go to the conveyor supplier with a real sample.